The answer doesn't make sense to me. I don't see why s should be fundamentally different than r. I think you should have a [tex]\delta(s+s')[/tex], with + rather than - because the arguments have opposite sign, and that there should be no integral over s.

But perhaps there is something about the definition of x(is,r) that you haven't told us that would change this ...

What kind of things might make s different to r in that way?
Perhaps these? "is" is wick rotated frequency; and started out as the fourier transform of time. r is real space. r is a vector and s is a scalar.